Nonpositive curvature: A geometrical approach to Hilbert–Schmidt operators

Abstract

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert–Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that G M , the Banach–Lie group generated by M, acts isometrically and transitively on M. Moreover, G M admits a polar decomposition relative to M, namely G M ≃ M × K as Hilbert manifolds (here K is the isotropy of p = 1 for the action I g : p ↦ g p g ∗ ), and also G M / K ≃ M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection Π M : Σ → M , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism N M ≃ Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed e a , we obtain e a = e x e v e x with e x ∈ M and v orthogonal to M at p = 1 . As a corollary we obtain decompositions for the full group of invertible elements G ≃ M × exp ( T 1 M ⊥ ) × K .